Inner ideals, compact tripotents and Čebyšëv subtriples of JB${}^{\ast }$-triples and C${}^{\ast }$-algebras

Abstract

The aim of this note is to study Čebyšëv JB${}^{\ast }$-subtriples of general JB${}^{\ast }$-triples. It is established that if $F$ is a non-zero Čebyšëv JB${}^{\ast }$-subtriple of a JB${}^{\ast }$-triple $E$, then exactly one of the following statements holds:\begin{enumerate}\item $F$ is a rank one JBW${}^{\ast }$-triple with dim$(F)\ge 2$ (i.e. a complex Hilbert space regarded as a type 1 Cartan factor). Moreover, $F$ may be a closed subspace of arbitrary dimension and $E$ may have arbitrary rank, \item $F=\mathbb{C}e$, where $e$ is a complete tripotent in $E$, \item $E$ and $F$ are rank two JBW${}^{\ast }$-triples, but $F$ may have arbitrary dimension, \item $F$ has rank greater or equal than three and $E=F$. \end{enumerate}

arXiv commentarXiv admin note: text overlap with arXiv:1412.3227